@misc{Koepf1995, author = {Koepf, Wolfram}, title = {Efficient Computation of Orthogonal Polynomials in Computer Algebra}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2080}, number = {SC-95-42}, year = {1995}, abstract = {Orthogonal polynomials \%like the Chebyshev polynomials can be calculated by computation of determinants, by the use of generating functions, in terms of Rodrigues formulas, by iterating recurrence equations, calculating the polynomial solutions of differential equations, through closed form representations and by other means. In this article, we give an overview about the efficiency of the above methods in Maple, Mathematica, and REDUCE. As a noncommercial package we include the MuPAD system.}, language = {en} } @misc{Koepf1995, author = {Koepf, Wolfram}, title = {Identities for Families of Orthogonal Polynomials and Special Functions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1675}, number = {SC-95-01}, year = {1995}, abstract = {In this article we present new results for families of orthogonal polynomials and special functions, that are determined by algorithmical approaches. In the first section, we present new results, especially for discrete families of orthogonal polynomials, obtained by an application of the celebrated Zeilberger algorithm. Next, we present algorithms for holonomic families \$f(n,x)\$ of special functions which possess a derivative rule. We call those families {\sl admissible}. A family \$f(n,x)\$ is holonomic if it satisfies a holonomic recurrence equation with respect to \$n\$, and a holonomic differential equation with respect to \$x\$, i.\ e. linear homogeneous equations with polynomial coefficients. The rather rigid property of admissibility has many interesting consequences, that can be used to generate and verify identities for these functions by linear algebra techniques. On the other hand, many families of special functions, in particular families of orthogonal polynomials, are admissible. We moreover present a method that generates the derivative rule from the holonomic representation of a holonomic family. \% whenever one exists. As examples, we find new identities for the Jacobi polynomials and for the Whittaker functions, and for families of discrete orthogonal polynomials by the given approach. Finally, we present representations for the parameter derivatives of the Gegenbauer and the generalized Laguerre polynomials.}, language = {en} } @misc{KoepfSchmersau1995, author = {Koepf, Wolfram and Schmersau, Dieter}, title = {On the De Branges Theorem}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1765}, number = {SC-95-10}, year = {1995}, abstract = {Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary proof of this statement based on the defining differential equations system rather than the closed representation of de Branges' function system. Our proof does neither use special functions (like Wilf's) nor the residue theorem (like Todorov's) nor the closed representation (like both), but is purely algebraic. On the other hand, by a similar algebraic treatment, the closed representation of de Branges' function system is derived. Our whole contribution can be looked at as the study of properties of the Koebe function. Therefore, in a very elementary manner it is shown that the known proofs of the Bieberbach and Milin conjectures can be understood as a consequence of the L{\"o}wner differential equation, plus properties of the Koebe function.}, language = {en} } @misc{Koepf1995, author = {Koepf, Wolfram}, title = {The Identification Problem for Transcendental Functions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1791}, number = {SC-95-13}, year = {1995}, abstract = {In this article algorithmic methods are presented that have essentially been introduced into computer algebra within the last decade. The main ideas are due to Stanley[1980] and Zeilberger[1990]. Some of them had already been discovered in the last century (see e.\ g.\ Beke[1894]), but because of the complexity of the underlying algorithms have fallen into oblivion. The combination of these ideas leads to a solution of the identification problem for a large class of transcendental functions. We present implementations of these algorithms in computer algebra systems.}, language = {en} } @misc{Koepf1995, author = {Koepf, Wolfram}, title = {REDUCE Packages on Power Series, Z-Transformation, Residues and Trigonometric Simplification}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5187}, number = {TR-95-03}, year = {1995}, abstract = {In this report, we present a collection of new REDUCE packages that recently have been developed. These are the packages {\tt FPS}, {\tt ZTRANS}, {\tt RESIDUE} and {\tt TRIGSIMP} on the following topics: \begin{itemize} \item[] FPS Calculation of differential equations and formal power series representations, in particular for orthogonal polynomials and special functions of the hypergeometric type. \item[] ZTRANS Calculation of the Z-Transformation and its inverse. \% to be used for the solution of linear recurrence equations. \item[] RESIDUE Calculation of residues of meromorphic functions. \item[] TRIGSIMP Simplification of expressions involving trigonometric and hyperbolic functions. \end{itemize} \% These packages can be obtained via anonymous {\tt ftp} from \% {\tt } in the directories {\tt ..}, ) For each of these packages, a description in form of a \LaTeX\ file is distributed together with the package. These documentations are collected here. In a final chapter, we show how to solve some difficult problems with these packages, and how they usefully can be combined.}, language = {en} }